ciuupi (version 1.1.0)

ciuupi: Compute the confidence interval that utilizes the uncertain prior information

Description

Compute the confidence interval that utilizes the uncertain prior information

Usage

ciuupi(alpha, a, c, x, bsvec, t, y, natural = 1, sig = NULL)

Arguments

alpha

1 - alpha is the minimum coverage probability of the confidence interval

a

A vector used to specify the parameter of interest

c

A vector used to specify the parameter about which we have uncertain prior information

x

The n by p design matrix

bsvec

The vector (b(1),...,b(5),s(0),...,s(5)) that specifies the CIUUPI

t

A number used to specify the parameter about which we have uncertain prior information

y

The n-vector of observed responses

natural

Equal to 1 (default) if b and s functions are obtained by natural cubic spline interpolation or 0 if obtained by clamped cubic spline interpolation

sig

Standard deviation of the random error. If a value is not specified then sig is estimated from the data.

Value

The confidence interval that utilizes uncertain prior information

Details

Suppose that $$y = X \beta + \epsilon$$ where \(y\) is a random \(n\)-vector of responses, \(X\) is a known \(n\) by \(p\) matrix with linearly independent columns, \(\beta\) is an unknown parameter \(p\)-vector and \(\epsilon\) with components that are iid normally distributed with zero mean and known variance. Then ciuupi will compute a confidence interval for \(\theta=\)a' \(\beta\) that utilizes the uncertain prior information that c' \(\beta\) - t = 0, where a and c are specified linearly independent vectors and t is a specified number.

In the example below we use the data set described in Table 7.5 of Box et al. (1963). A description of the parameter of interest and the parameter about which we have uncertain prior information is given in Dicsussion 5.8, p.3426 of Kabaila and Giri (2009).

References

Box, G.E.P., Connor, L.R., Cousins, W.R., Davies, O.L., Hinsworth, F.R., Sillitto, G.P. (1963) The Design and Analysis of Industrial Experiments, 2nd edition, reprinted. Oliver and Boyd, London.

Kabaila, P. and Giri, K. (2009) Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419 - 3429.

Examples

# NOT RUN {
# Specify alpha, a, c, x
alpha <- 0.05
a <- c(0, 2, 0, -2)
c <- c(0, 0, 0, 1)
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
x <- cbind(rep(1, 4), x1, x2, x1*x2)

# Find the vector (b(1),b(2),...,b(5),s(0),s(1),...,s(5)) that specifies the
# CIUUPI: (this may take a few minutes to run)
# }
# NOT RUN {
bsvec <- bsciuupi(alpha, a = a, c = c, x = x)
# }
# NOT RUN {
# The result (to 7 decimal places) is
bsvec <- c(-0.03639701, -0.18051953, -0.25111411, -0.15830362, -0.04479113,
           1.71997203, 1.79147968, 2.03881195, 2.19926399, 2.11845381,
           2.00482563)

# Specify t and y
t <- 0
y <- c(87.2, 88.4, 86.7, 89.2)

# Find the CIUUPI
res <- ciuupi(alpha, a, c, x, bsvec, t, y, natural = 1, sig = 0.8)
res

# }